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                        <h1 class="single-title flipInX">图像分割(Segmentation)——K-Means, 最小割, 归一化图割</h1><div class="post-meta summary-post-meta"><span class="post-category meta-item">
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                            <span>目录</span>
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                        <div class="details-content toc-content" id="toc-content-static"><nav id="TableOfContents">
  <ul>
    <li><a href="#gestalt理论">Gestalt理论</a></li>
    <li><a href="#k-means聚类">K-Means聚类</a></li>
    <li><a href="#mean-shift-聚类">Mean shift 聚类</a></li>
    <li><a href="#image-as-graphs">Image as Graphs</a></li>
  </ul>
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                    </div><p>图像分割是将图片将相似的部分分割成相同的块</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731140124510.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731140124510.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<h2 id="gestalt理论" class="headerLink"><a href="#gestalt%e7%90%86%e8%ae%ba" class="header-mark"></a>Gestalt理论</h2><p>解释物体分割的底层原理
将同一个东西群组在一起,集合中的元素可以具有由关系产生的属性</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731141124328.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731141124328.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>Gestalt中常见的一些分组的情况</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731142105698.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731142105698.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>现实生活中的分组现象</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731142812410.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731142812410.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure>
<strong>将这种思想转化为算法</strong></p>
<h2 id="k-means聚类" class="headerLink"><a href="#k-means%e8%81%9a%e7%b1%bb" class="header-mark"></a>K-Means聚类</h2><p><strong>主要思想</strong>：相似的像素应该属于同一类</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/2020073114304895.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/2020073114304895.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure>
<strong>像素表达</strong>：每个像素可以使用一个多维向量来表示，如(R, G, B)的三维向量，(R, G, B, x, y)的五维向量</p>
<blockquote>
<p><strong><a href="https://www.cnblogs.com/pinard/p/6164214.html" target="_blank" rel="noopener noreffer">K-Means算法</a></strong>：
“物以类聚、人以群分”：</p>
<ol>
<li>
<p>首先输入k的值，即我们希望将数据集经过聚类得到k个分组。</p>
</li>
<li>
<p>从数据集中随机选择k个数据点作为初始大哥（质心，Centroid）</p>
</li>
<li>
<p>对集合中每一个小弟，计算与每一个大哥的距离（距离的含义后面会讲），离哪个大哥距离近，就跟定哪个大哥。</p>
</li>
<li>
<p>这时每一个大哥手下都聚集了一票小弟，这时候召开人民代表大会，每一群选出新的大哥（其实是通&gt;过算法选出新的质心）。</p>
</li>
<li>
<p>如果新大哥和老大哥之间的距离小于某一个设置的阈值（表示重新计算的质心的位置变化不大，趋于&gt;稳定，或者说收敛），可以认为我们进行的聚类已经达到期望的结果，算法终止。</p>
</li>
<li>
<p>如果新大哥和老大哥距离变化很大，需要迭代3~5步骤。</p>
</li>
</ol>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731145629120.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731145629120.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
</blockquote>
<p>基于灰度值或颜色的K-means聚类本质上是对图像属性的矢量量化</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731143822837.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731143822837.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<ul>
<li>语义分割：将不同位置上同样的物品分成同一类（如上图三）使用(R, G, B)的三维向量描述像素</li>
<li>实列分割：将不同位置上同样的物品分成不同类，此时就需要考虑像素位置对分割的影响，使用(R, G, B, x, y)的五维向量描述像素</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731150331930.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731150331930.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<h2 id="mean-shift-聚类" class="headerLink"><a href="#mean-shift-%e8%81%9a%e7%b1%bb" class="header-mark"></a>Mean shift 聚类</h2><p>K-Means算法可以很简单的进行分割，但是初始值对结果的影响非常大，<strong>Mean Shift</strong> 算法处理时将像素密度的峰值作为初始值，在特征空间中寻找密度的模态或局部最大值</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731152036698.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731152036698.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>所有轨迹通向相同模态的区域





<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731151831291.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731151831291.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<blockquote>
<p><strong><a href="https://blog.csdn.net/csdnforyou/article/details/81161840" target="_blank" rel="noopener noreffer">Mean Shift</a></strong>：</p>
<ol>
<li>定义特征值：(color, gradients, texture, etc)</li>
<li>在单个特征点上初始化窗口</li>
<li>对每个窗口执行Mean Shift直到收敛</li>
<li>合并接近相同“峰值”或模式的窗口
<br></li>
</ol>
<p><strong>分割实例</strong>：</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731153417647.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731153417647.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure>
<strong>优势</strong>：</p>
<ul>
<li>算法计算量不大，在目标区域已知的情况下完全可以做到实时跟踪；</li>
<li>采用核函数直方图模型，对边缘遮挡、目标旋转、变形和背景运动不敏感。</li>
</ul>
<p><strong>缺点</strong>:</p>
<ul>
<li>缺乏必要的模板更新；</li>
<li>跟踪过程中由于窗口宽度大小保持不变，当目标尺度有所变化时，跟踪就会失败；</li>
<li>当目标速度较快时，跟踪效果不好；</li>
<li>直方图特征在目标颜色特征描述方面略显匮乏，缺少空间信息。</li>
</ul>
</blockquote>
<h2 id="image-as-graphs" class="headerLink"><a href="#image-as-graphs" class="header-mark"></a>Image as Graphs</h2><ul>
<li>节点代表像素</li>
<li>每对节点之间代表边缘(或每对“足够接近”的像素)</li>
<li>每条边都根据两个节点的亲和力或相似性进行加权</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/2020073115523359.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/2020073115523359.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<ul>
<li>找一条边将，将像素之间的联系切断</li>
<li>将切断的边的权重相加，使和最小（相似的像素应该在相同的部分，不同的像素应该在不同的部分）</li>
</ul>
<p><strong>定义权值</strong>
$$
exp(-\frac{1}{2\sigma^2}dist(X_i,X_j)^2)
$$</p>
<ol>
<li>
<p>假设用特征向量x表示每个像素，并定义一个适合于该特征表示的距离函数: $dist(X_i,X_j)^2$</p>
</li>
<li>
<p>然后利用广义高斯核将两个特征向量之间的距离转化为相似性</p>
</li>
<li>
<p>小的$\sigma$：仅群聚邻近点；大的$\sigma$：群聚距离远的点</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731160907845.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731160907845.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
</li>
</ol>
<p><strong>最小割</strong>：将权值小的边去掉</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731164000819.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731164000819.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>最小割中趋向于将点单独的割出来，就会导致分割出很多“小块”</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731164409976.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731164409976.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p><strong>归一化图割</strong></p>
<blockquote>
<p>$$
\frac{W(A,B)}{W(A,V)}+\frac{W(A,B)}{W(B,V)}
$$</p>
<ul>
<li>$W(A,B)$：表示集合$A, B$中所有边的权重的和</li>
<li>$W(A,V)，W(B,V)$：分别表示分割后集合$A, B$内边的个数，当总的边数一定，其中一个$W$很大时，另一个就会很小，导致整个算式所计算的权重就会很大</li>
</ul>
</blockquote>
<ul>
<li>
<p>定义一个邻接关系矩阵$W$，$(i,j)$表示第i个像素与第j个像素之间的权重，是一个对称的矩阵</p>
</li>
<li>
<p>定义矩阵$D$，是一个对角的矩阵$D_{ii} = \Sigma_jW(i, j)$</p>
</li>
</ul>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/2020073117251281.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/2020073117251281.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>$$
\frac{y^T(D-W)y}{y^TDy}
$$</p>
<ul>
<li>
<p>归一化图割权重计算：其中，y是指示向量，如果第i个特征点属于A，那么它在第i个位置的值应该为1，反之则为0；利用拉格朗日算法求权重<strong>第二小</strong>（最小的是0）的特征值对应的特征向量y：$(D-W)y=\lambda y$
所求出的y不是[0,…,1,0,0…]而是存在小数的向量，需要设置一个门限，将大于门限的设置为1，小于门限的设置为0</p>
</li>
<li>
<p>使用特征向量y的项对图进行二分割</p>
</li>
<li>
<p>改变门限，迭代将图像分成多个区块，也可以使用K-Means将得到的特征向量分类，达到分割多块的效果</p>
</li>
</ul>
<p>分割效果：</p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731175641209.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731175641209.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p>




<figure class="render-image"><a target="_blank" href="https://img-blog.csdnimg.cn/20200731175536315.png" title=" " >
        <img loading="lazy" decoding="async"
             class="render-image"
             src="https://img-blog.csdnimg.cn/20200731175536315.png"
            alt=" "
        />
    </a><figcaption class="image-caption"> </figcaption>
</figure></p>
<p><strong>学习资源：<a href="https://www.bilibili.com/video/BV1nz4y197Qv" target="_blank" rel="noopener noreffer">北京邮电大学计算机视觉——鲁鹏</a></strong></p>
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